Is there a Whitney sum formula for topological rational Pontryagin classes? I thought the answer is yes, but now I cannot find a reference. Is it even true? The PL case would also be of interest.
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asked May 22, 2020 at 2:07
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1$\begingroup$ Let p(E) denote the total (integral) Pontryagin class of a real bundle E. It's not too hard to show that p(E (+) F) = p(E) * p(F) mod 2-torsion. The 2-torsion term is stated in Theorem 1.6 of Brown's "The Cohomology of BSO_n and BO_n with Integer Coefficients". $\endgroup$– skdCommented May 22, 2020 at 4:23
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1$\begingroup$ @sdk: Brown's paper is about vector bundles, while my question is about locally trivial bundles with fiber homeomorphic to $\mathbb R^n$. $\endgroup$– Igor BelegradekCommented May 22, 2020 at 12:40
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$\begingroup$ Ah, yes. Sorry, didn't read carefully enough. $\endgroup$– skdCommented May 22, 2020 at 17:29
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1 Answer
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Yes. A simple argument is that $BO \to BTOP$ is a rational equivalence and an H-space map (in fact even an infinite loop map), so it follows from the Whitney sum formula for vector bundles.
Edit: The argument for this is as follows. Let $\mu : BTOP \times BTOP \to BTOP$ be the map corresponding to Whitney sum of (stable, topological) bundles. The question is whether the identity $$\mu^* p_n = \sum_{ a + b = n} p_a \otimes p_b$$ holds. As the map $BO \to BTOP$ is a rational equivalence and an H-space map, it is equivalent to verify this equation in the cohomology of $BO$ instead, where it indeed holds.
answered May 22, 2020 at 7:05
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1$\begingroup$ Thank you! For my record here is a reference: on p.215 of Boardman-Vogt's book "Homotopy invariant algebraic structures on topological spaces" the authors say (roughtly) that since $O\to Top\to F$ are topological monoid homomorphisms under the Whitney sum, these maps are infinite loop space maps, and hence so are the corresponding maps of classifying spaces $BO\to BTop\to BF$. Actually, instead of this last sequence they write $BO\to Top\to F$ which I assume is a misprint. $\endgroup$ Commented May 22, 2020 at 12:29
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$\begingroup$ Is it true more generally that for any topological monoid (or at least group) homomorphism $G\to H$ the corresponding map of classifying spaces $BG\to BH$ is an infinite loop map? $\endgroup$ Commented May 22, 2020 at 12:29
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$\begingroup$ @ConnorMalin: thank you, I see now that Boardman-Vogt's result has quite a few technical assumptions; they speak of symmetrical monoidal functors. I gather, the point is that Whitney sum is commutative. $\endgroup$ Commented May 22, 2020 at 13:05
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$\begingroup$ Sorry, I deleted my comment in fear that I was missing something. For posterity, I said that usually a group will not be an infinite loop space. $\endgroup$ Commented May 22, 2020 at 13:35
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1$\begingroup$ $[X,BSO_\mathbb{Q}]\rightarrow [X,BSTOP_\mathbb{Q}]$ is an isomorphism. The rational Pontryagin classes of $x\in [X,BSTOP]$ only depend on its image in $[X,BSTOP_\mathbb{Q}]$. $\endgroup$ Commented May 23, 2020 at 12:52